Remarks on the relative isoperimetric profile of polygonal domains in $\mathbb{R}^2$

Ezra Nance; Jason DeVito; Robert DeYeso III; Robert Niedzialomski

arxiv: 2605.23149 · v1 · pith:L5D6XVIQnew · submitted 2026-05-22 · 🧮 math.DG

Remarks on the relative isoperimetric profile of polygonal domains in mathbb{R}²

Jason DeVito , Robert DeYeso III , Ezra Nance , Robert Niedzialomski This is my paper

Pith reviewed 2026-05-25 03:34 UTC · model grok-4.3

classification 🧮 math.DG

keywords relative isoperimetric problempolygonal domainscornersisoperimetric profilesquare with corner removedR^2differential geometry

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The pith

The relative isoperimetric problem is solved for a square with a square corner removed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops techniques for solving the relative isoperimetric problem on polygonal domains in the plane, with special attention to corners. It applies these techniques to completely solve the problem for a square domain with one square corner removed. A sympathetic reader would care because the relative isoperimetric problem asks for the shortest boundary length that encloses a prescribed area inside a given domain, and corners create points where usual smoothness assumptions break down.

Core claim

We develop techniques for solving the relative isoperimetric problem on polygonal domains in R^2, with special attention paid to corners. As an application, we solve the relative isoperimetric problem for a square with a square corner removed.

What carries the argument

Techniques for handling corners when solving the relative isoperimetric problem on polygonal domains.

Load-bearing premise

The techniques developed for handling corners in polygonal domains are sufficient to produce a complete and rigorous solution without additional regularity assumptions that might fail at vertices.

What would settle it

An explicit curve inside the square with corner removed that encloses a given area using strictly less relative perimeter than the profile claimed in the solution.

Figures

Figures reproduced from arXiv: 2605.23149 by Ezra Nance, Jason DeVito, Robert DeYeso III, Robert Niedzialomski.

**Figure 1.** Figure 1: Region S1 is a quarter circle hitting both boundary components of Qa perpendicularly. Regions S2 and S3 are bounded by line segments parallel to the sides of Qa, and the boundary of Region S4 is a portion of a circle connecting the non-convex corner to a side of length 1. Acknowledgements The first author was supported by the NSF through DMS-2405266. He is grateful for the support. 2 [PITH_FULL_IMAGE:figu… view at source ↗

**Figure 2.** Figure 2: Two examples showing that the boundary of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Neighborhood of a corner K with a circular arc. By Proposition 2.2, ∂S has only finitely many smooth boundary components. Therefore, we may find a neighborhood of K, which intersects ∂S only in a portion of this circular arc. Given any point T on this circular arc, we let T ′ denote the intersection of the tangent line at T and Q1, see [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: On the left we have ∂S intersecting a corner K. On the right we have a deformation ∂S′ which avoids corner K. We will now deform ∂S to ∂S′ with the desired properties. Consider a point A which is ε units away from the corner, and let B be the point directly above A so that |S| = |S ′ |. Let C be the intersection point of KP and AB, and let w be the length of BC. Note that in order for |S| = |S ′ |, we need… view at source ↗

**Figure 5.** Figure 5: Both possibilities for a circular arc to touch a corner acutely. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Two smooth boundary components which are tangent at [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: One minimizing region We define four regions S1, S2, S3, and S4 as follows. The region S1 is any region bounded by a quarter circle ending on sides of Qa orthogonally, except that we do not allow a quarter circle surrounding the non-convex corner. The region S2 is bounded by a line segment of length 1. The region S3 is bounded by a line segment of length 1 − a. And finally, the region S4 is bounded by a cu… view at source ↗

**Figure 8.** Figure 8: Reflecting the region We now work towards completing the proof of Theorem 4.3 by determining for each a ∈ [0, 1) and each t ∈ [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗

read the original abstract

We develop techniques for solving the relative isoperimetric problem on polygonal domains in $\mathbb{R}^2$, with special attention paid to corners. As an application, we solve the relative isoperimetric problem for a square with a square corner removed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They develop corner techniques for the relative isoperimetric problem and solve it explicitly for a square minus one corner, but the vertex regularity needs direct verification.

read the letter

The main thing here is that the authors develop techniques for the relative isoperimetric problem on polygonal domains with special attention to corners and then apply them to solve the problem for a square with a square corner removed. This gives an explicit profile for that domain, which the abstract presents as new. The work extends standard variational methods by tailoring them to handle the angles at corners, and the application to this specific example is the concrete advance. It is useful because explicit solutions in domains with corners are not common, and this one illustrates behavior near the vertices. The paper does a reasonable job of identifying where the usual smooth-domain arguments break and trying to fix them locally at the corners. The soft spot is the one the stress-test flags. The claim that the techniques close the problem rests on whether the constructed minimizer satisfies the first-variation and regularity conditions at every vertex, including the new 90-degree corners created by the cut. If the full derivations show this without extra curvature assumptions that fail at those points, the result is fine; if the argument uses interior regularity or approximation that does not carry over to the vertices, the solution for the domain is incomplete. The abstract does not reveal circularity or free parameters, and the citation pattern looks like standard references to prior isoperimetric work. This paper is for people working on isoperimetric problems in non-smooth or polygonal domains. A reader who wants explicit examples or methods for handling corners will get value from the techniques and the solved case. It deserves a serious referee because the claim is specific, the example is new, and the proofs can be checked in detail. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript develops techniques for the relative isoperimetric problem on polygonal domains in R^2, paying special attention to corners, and applies these techniques to solve the problem explicitly for the domain consisting of a square with one square corner removed.

Significance. If the corner techniques are fully rigorous, the work supplies an explicit solution to a concrete relative isoperimetric problem on a non-convex polygonal domain. This is a modest but useful contribution to the literature on isoperimetric problems in domains with corners, where explicit profiles are rarely available.

major comments (1)

[Application section (the square-with-corner-removed case)] The central claim that the developed corner techniques produce a complete, rigorous solution for the square-with-corner-removed domain rests on the assertion that all first-variation and regularity conditions hold at every vertex (including the newly created ones). Without an explicit verification that the constructed candidate satisfies the necessary angle or curvature conditions at 90-degree corners without hidden interior-regularity assumptions, the solution remains formally incomplete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the application section. We address it below.

read point-by-point responses

Referee: [Application section (the square-with-corner-removed case)] The central claim that the developed corner techniques produce a complete, rigorous solution for the square-with-corner-removed domain rests on the assertion that all first-variation and regularity conditions hold at every vertex (including the newly created ones). Without an explicit verification that the constructed candidate satisfies the necessary angle or curvature conditions at 90-degree corners without hidden interior-regularity assumptions, the solution remains formally incomplete.

Authors: We agree that an explicit verification of the first-variation and regularity conditions at all vertices (including the 90-degree corners and newly created ones) would strengthen the rigor of the application. In the revised manuscript we will add a short dedicated paragraph (or subsection) that directly checks these conditions for the constructed candidate, confirming that the angle conditions hold without hidden interior assumptions. This addresses the concern and renders the solution formally complete. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on variational techniques without reduction to inputs

full rationale

The paper develops techniques for the relative isoperimetric problem on polygonal domains with attention to corners and applies them to solve the problem for a square with one corner removed. No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the central claims rest on standard first-variation and regularity analysis that is presented as independent of the target result. The derivation chain is self-contained against external benchmarks in geometric measure theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard results from the calculus of variations and geometric measure theory for curves in the Euclidean plane; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of rectifiable curves and perimeter functionals in R^2 hold at polygonal corners.
The relative isoperimetric problem is posed in the classical setting of Euclidean geometry and calculus of variations.

pith-pipeline@v0.9.0 · 5567 in / 1012 out tokens · 93544 ms · 2026-05-25T03:34:18.843144+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: If K convex then ∂S does not contain K; if non-convex then at most one smooth component contains K and makes no acute angle.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Explicit f_a via quarter-circles, line segments, and S4 circular arcs meeting non-convex corner (eq. 4.2, Lemma 4.1).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Brezis and A

H. Brezis and A. Bruckstein. A sharp relative isoperimetric inequality for the square.C. R. Math. Acad. Sci. Paris, 359(9):1191–1199, 2021

work page 2021
[2]

J. Choe, M. Ghomi, and M. Ritor´ e. The relative isoperimetric inequality outside a convex domain inR n.Calc. Var. PDE, 29(4):421–429, 2007

work page 2007
[3]

Gonzalez, U

E. Gonzalez, U. Massari, and I. Tamanini. On the regularity of boundaries of sets minimizing perimeter with a volume constraint.Indiana University Mathematics Journal, 32(1):25–37, 1983

work page 1983
[4]

Gr¨ uter

M. Gr¨ uter. Boundary regularity for solutions of a partitioning problem. Archive for Rational Mechanics and Analysis, 97:261–270, 1987

work page 1987
[5]

M¨ ader-Baumdicker, R

E. M¨ ader-Baumdicker, R. Neumayer, J. Park, and M. Rupflin. Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane.arXiv2508.21198, 2025

work page arXiv 2025
[6]

Stredulinsky and W.P

E. Stredulinsky and W.P. Ziemer. Area minimizing sets subject to a volume constraint in a convex set.J. Geom. Anal., 7:653–677, 1997. 24

work page 1997

[1] [1]

Brezis and A

H. Brezis and A. Bruckstein. A sharp relative isoperimetric inequality for the square.C. R. Math. Acad. Sci. Paris, 359(9):1191–1199, 2021

work page 2021

[2] [2]

J. Choe, M. Ghomi, and M. Ritor´ e. The relative isoperimetric inequality outside a convex domain inR n.Calc. Var. PDE, 29(4):421–429, 2007

work page 2007

[3] [3]

Gonzalez, U

E. Gonzalez, U. Massari, and I. Tamanini. On the regularity of boundaries of sets minimizing perimeter with a volume constraint.Indiana University Mathematics Journal, 32(1):25–37, 1983

work page 1983

[4] [4]

Gr¨ uter

M. Gr¨ uter. Boundary regularity for solutions of a partitioning problem. Archive for Rational Mechanics and Analysis, 97:261–270, 1987

work page 1987

[5] [5]

M¨ ader-Baumdicker, R

E. M¨ ader-Baumdicker, R. Neumayer, J. Park, and M. Rupflin. Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane.arXiv2508.21198, 2025

work page arXiv 2025

[6] [6]

Stredulinsky and W.P

E. Stredulinsky and W.P. Ziemer. Area minimizing sets subject to a volume constraint in a convex set.J. Geom. Anal., 7:653–677, 1997. 24

work page 1997